In spherical polar coordinates, the continuity equations for density, momentum, and energy for a mono-atomic gas along the radial direction are respectively [1]}, [2]}, [3]},
\(&&\partial _t\rho +\frac{1}{r^{d-1}} \partial _r(r^{d-1}\rho u) = 0,\\&&\partial _t(\rho u) + \frac{1}{r^{d-1}}\partial _r(r^{d-1}\rho u^2 )+ \partial _r p = \frac{1}{r^{d-1}}\partial _r(2\mu r^{d-1}\partial _r u) - \frac{2\mu (d-1)u}{r^2}, \\&&\partial _t\Big [\frac{1}{2}\rho u^2 + \frac{1}{\gamma -1}\rho T\Big ] + \frac{1}{r^{d-1}}\partial _r\Big [r ^{d-1}\Big (\frac{1}{2}\rho u^2 + \frac{1}{\gamma -1}\rho T + p \Big )u\Big ] \nonumber \\&&\qquad = \frac{1}{r^{d-1}}\partial _r(2\mu r^{d-1} u\partial _r u) + \frac{1}{r^{d-1}}\partial _r( r ^{d-1} \lambda \partial _r T) ,\) 
